In this thesis, the diffusive limit of active particle motion in R^d is studied via a technique based on homogenisation. Thereafter, this study is extended to active particle motion on a Riemannian manifold.

Furthermore, as an application of active particle motion, a connection is made with the Dirac equation. On the basis of this connection, a Monte Carlo method is developed to find the ground state of a Dirac equation with static potential. The core idea of this method is based on the Diffusion Monte Carlo method for the Schrödinger equation.